Excited states in DFT

The ADFT/ASCF-DFT scheme has been met with considerable reservation. Thus, ADFT/ASCF-DFT assumes implicitly that a transition can be represented by an excitation involving only two orbitals, an assumption that seems not generally to be satisfied. Also, the variational optimization in ASCF-DFT of the orbitals makes it difficult to ensure orthogonality between different excited state determinants when many transitions are considered, resulting ultimately in a variational collapse. Finally, it has been questioned [110] whether there exists a variational principle for excited states in DFT. In spite of this, some of the first pioneering chemical applications of DFT involved ASCF-DFT calculations on excitation energies [36, 113-116] for transition metal complexes and ASCF-DFT is still widely used [117-121]. 

Several more rigorous approaches to excited states in DFT, which do not require the KS eigenvalues to have physical meaning, are mentioned in Sec. 6. 

In the remainder of this chapter, I first review the status of the treatment of excited states in DFT and then go into a detailed analysis of where and how fimctionals can be improved. 

A systematic approach to excited states in DFT is ensemble DFT, developed by Theophilou and further elaborated by Oliveira et al. " In this formalism the functional depends on the particular choice for the ensemble, and a simple approximation for this dependence is available. " Some applications of this method have been worked out by Nagy and a recent analysis of this method is presented in Ref. [96]. 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Most of the results presented in Table 9 do not take into account selfenergy corrections, which are necessary in order to describe, in a proper way, the one-particle excited states. In the fourth column of Table 9 we report the GW corrected band-gaps, for the smallest Ge-NWs in the [111],[110] directions, and for all the [100] Ge-NW. A complete discussion on this part can be found elsewhere [121], We can see (Table 9, fifth column) that the effect of the GW correction is an opening of the DFT-LDA gap, by an amount which... 

Fig. 4 Energy correlation diagram between the gas phase (/G) and solvated (toluene) (/S) lowest excited states in a TFB F8BT model system calculated by TD-DFT (B3LYP/6-31G(d)). The eclipsed vs. staggered structures as shown in Fig. 3 are compared. The lowest-lying excitonic (XT) and charge transfer (CT) states are highlighted in red. Solvation effects tend to stabilize the CT state. Reprinted with permission from Ref. [41]. Copyright 2007, American Institute of Physics.

Some other polymers of the same type with valence (I) were also prepared (Fig. 17). They exhibit almost the same structure, except that halides are replaced by diphosphine ligands (diphos) such as bis(diphenylphosphino) butane (dppb), bis(diphenylphosphino)pentane (dpppen), and bis(diphenyl-phosphino)hexane (dpph).36,40 Again a model complex, compound 25, was studied as reference (Fig. 17). The electronic spectra exhibit an absorption band near 480 nm. These coordination materials are not luminescent at room temperature but are luminescent in solution in butyronitrile at low temperature (i.e., 77 K). Density functional theory (DFT) calculations showed that luminescence arises from a da-da triplet excited state. In these polymers, the nature of the phosphine ligand has a crucial effect on absorption and emission bands. Such behavior is explained by the increase in electronic density on the... 

Density functional theory (DFT) calculations were also carried out to assign the molecular orbitals involved in the transitions that lead to luminescence, concluding that metal centered (du )1(pu)1 or (da )1 (pa)1 excited states are responsible for the luminescence in the solid state, while in dilute solutions the luminescence arises from ira excited states in the pentafluorophenyl ligands or from ir-MMCT transitions. 

G. Scalmani, M. J. Frisch, B. Mennucci, J. Tomasi, R. Cammi and V. Barone, Geometries and properties of excited states in the gas phase and in solution. Theory and application of a time-dependent DFT polarizable continuum model, J. Chem. Phys., 124 (2006) 094107. 

Vlcek Jr. A, Zalis S. Modeling of charge-transfer transitions and excited states in d6 transition metal complexes by DFT techniques. Coord Chem Rev 2007 251 258-87. 

DFT-like approximation techniques might provide a Fock-like Agff operator that can incorporate static and dynamic correlation effects of excited states in a mean-field sense. 

Consequently, DFT is restricted to ground-state properties. For example, band gaps of semiconductors are notoriously underestimated [142] because they are related to the properties of excited states. Nonetheless, DFT-inspired techniques which also deal with excited states have been developed. These either go by the name of time-dependent density-functional theory (TD-DFT), often for molecular properties [147], or are performed in the context of many-body perturbation theory for solids such as Hedin s GW approximation [148]. 

Several approaches for calculating excited states in protein environments were proposed to improve the ordinary QM/MM description. The effect of polarization was included as a classical force field [27], and the excitation energy calculated for bacteriorhodopsin (bR) was 0.34 eV less than that from a fixed-charge non-polarizable QM/MM method [27]. Later, a triple-layer QM1/QM2/MM approach was proposed, and DFT(PBEO) calculations were performed for the QM2 layer, which consisted of the amino acids 4 A from the retinal PSB [28]. The calculated excitation energy of bR was only 0.08 eV smaller than that obtained using the ordinary QM/MM method [28]. In another study, an empirical polarization model combined with the QM/MM calculation produced a red shift of 0.14-0.17 eV [29]. However, these pioneering studies neglected the CT effects between the retinal and the protein environments. 

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